Treat the cause. Antibiotics are only good for treating some of the bacteria, but are not able to remove all of the bugs, so they regrow. You will need to find out why you have a tooth or gum infection and then treat the cause. If the problem is inside the tooth a root canal may be needed. If it is a gum problem you might try something like the perio protect method.
1-2 Days. Traditionally antibiotics can take anywhere from 1-2 days to start working on the infection, depending on the severity of the infection. I tell my patients if after 2 days, the antibiotic doesn't seem to be working, let us know and we can change the antibiotic.
About 24hrs. The problem is not the antibiotic but the suseptablility of the bacteria to the antibiotic. It takes time for the antibiotic to build up in the blood to where it affects the bacteria. Dental infections often need to be opened or "drained" so the antibiotic can actually get to the infection.